L(s) = 1 | − 4·11-s − 25-s − 2·49-s − 4·59-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
L(s) = 1 | − 4·11-s − 25-s − 2·49-s − 4·59-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4488530674\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4488530674\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$ | \( ( 1 + T )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.228213744599735915277949965899, −8.475054487925109906245152847210, −8.375646719949536093022274072792, −7.949759081703459017712212768714, −7.64280772688851557876122890586, −7.50341078341659257465136480910, −7.07518669611979680886792163217, −6.26646123443303154053973883447, −6.13492548583102592963685286850, −5.59410818043879712713621234199, −5.31811541550600676038158744916, −4.77873223941865013788183837483, −4.76319198064855138069403366956, −4.11531973979706627298228323569, −3.29257989099249375954507285473, −2.94774358625620408474538972010, −2.82405056728476985418503660155, −1.99146617552321603127566000554, −1.78842535586474670052568472846, −0.38898812579356765423182752513,
0.38898812579356765423182752513, 1.78842535586474670052568472846, 1.99146617552321603127566000554, 2.82405056728476985418503660155, 2.94774358625620408474538972010, 3.29257989099249375954507285473, 4.11531973979706627298228323569, 4.76319198064855138069403366956, 4.77873223941865013788183837483, 5.31811541550600676038158744916, 5.59410818043879712713621234199, 6.13492548583102592963685286850, 6.26646123443303154053973883447, 7.07518669611979680886792163217, 7.50341078341659257465136480910, 7.64280772688851557876122890586, 7.949759081703459017712212768714, 8.375646719949536093022274072792, 8.475054487925109906245152847210, 9.228213744599735915277949965899